Week 10 Day 3 Homework
MVP
Question 1
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library(fastDummies)
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## na.warnhouses <- read_csv("data/kc_house_data.csv")## Rows: 21613 Columns: 21
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): id
## dbl (19): price, bedrooms, bathrooms, sqft_living, sqft_lot, floors, waterf...
## dttm (1): date
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.summary(houses)## id date price
## Length:21613 Min. :2014-05-02 00:00:00.00 Min. : 75000
## Class :character 1st Qu.:2014-07-22 00:00:00.00 1st Qu.: 321950
## Mode :character Median :2014-10-16 00:00:00.00 Median : 450000
## Mean :2014-10-29 04:38:01.96 Mean : 540088
## 3rd Qu.:2015-02-17 00:00:00.00 3rd Qu.: 645000
## Max. :2015-05-27 00:00:00.00 Max. :7700000
## bedrooms bathrooms sqft_living sqft_lot
## Min. : 0.000 Min. :0.000 Min. : 290 Min. : 520
## 1st Qu.: 3.000 1st Qu.:1.750 1st Qu.: 1427 1st Qu.: 5040
## Median : 3.000 Median :2.250 Median : 1910 Median : 7618
## Mean : 3.371 Mean :2.115 Mean : 2080 Mean : 15107
## 3rd Qu.: 4.000 3rd Qu.:2.500 3rd Qu.: 2550 3rd Qu.: 10688
## Max. :33.000 Max. :8.000 Max. :13540 Max. :1651359
## floors waterfront view condition
## Min. :1.000 Min. :0.000000 Min. :0.0000 Min. :1.000
## 1st Qu.:1.000 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:3.000
## Median :1.500 Median :0.000000 Median :0.0000 Median :3.000
## Mean :1.494 Mean :0.007542 Mean :0.2343 Mean :3.409
## 3rd Qu.:2.000 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:4.000
## Max. :3.500 Max. :1.000000 Max. :4.0000 Max. :5.000
## grade sqft_above sqft_basement yr_built
## Min. : 1.000 Min. : 290 Min. : 0.0 Min. :1900
## 1st Qu.: 7.000 1st Qu.:1190 1st Qu.: 0.0 1st Qu.:1951
## Median : 7.000 Median :1560 Median : 0.0 Median :1975
## Mean : 7.657 Mean :1788 Mean : 291.5 Mean :1971
## 3rd Qu.: 8.000 3rd Qu.:2210 3rd Qu.: 560.0 3rd Qu.:1997
## Max. :13.000 Max. :9410 Max. :4820.0 Max. :2015
## yr_renovated zipcode lat long
## Min. : 0.0 Min. :98001 Min. :47.16 Min. :-122.5
## 1st Qu.: 0.0 1st Qu.:98033 1st Qu.:47.47 1st Qu.:-122.3
## Median : 0.0 Median :98065 Median :47.57 Median :-122.2
## Mean : 84.4 Mean :98078 Mean :47.56 Mean :-122.2
## 3rd Qu.: 0.0 3rd Qu.:98118 3rd Qu.:47.68 3rd Qu.:-122.1
## Max. :2015.0 Max. :98199 Max. :47.78 Max. :-121.3
## sqft_living15 sqft_lot15
## Min. : 399 Min. : 651
## 1st Qu.:1490 1st Qu.: 5100
## Median :1840 Median : 7620
## Mean :1987 Mean : 12768
## 3rd Qu.:2360 3rd Qu.: 10083
## Max. :6210 Max. :871200No missing data
# removing columns that aren't needed
houses_tidy <- houses %>%
select(-c(date, id, sqft_living15, sqft_lot15, zipcode)) %>%
# converting waterfront column to logical
mutate(waterfront = as.logical(waterfront)) %>%
# converting yr_renovated to logical renovated column
mutate(yr_renovated = ifelse(yr_renovated == 0, FALSE, TRUE)) %>%
rename(renovated = yr_renovated) %>%
# convert view to factor (categorical ordinal)
mutate(view = factor(view, levels = c(0, 1, 2, 3, 4))) %>%
# convert condition to factor (categorical ordinal)
mutate(condition = factor(condition, levels = c(1, 2, 3, 4, 5))) %>%
# lets group together grade - low (1-3), average (4-10), high (11-13)
mutate(grade = case_when(
grade < 4 ~ "low",
grade >10 ~ "high",
TRUE ~ "average"
),
grade = factor(grade, levels = c("low", "average", "high"))
)Question 2
alias(lm(price ~ ., data = houses_tidy))## Model :
## price ~ bedrooms + bathrooms + sqft_living + sqft_lot + floors +
## waterfront + view + condition + grade + sqft_above + sqft_basement +
## yr_built + renovated + lat + long
##
## Complete :
## (Intercept) bedrooms bathrooms sqft_living sqft_lot floors
## sqft_basement 0 0 0 1 0 0
## waterfrontTRUE view1 view2 view3 view4 condition2 condition3
## sqft_basement 0 0 0 0 0 0 0
## condition4 condition5 gradeaverage gradehigh sqft_above yr_built
## sqft_basement 0 0 0 0 -1 0
## renovatedTRUE lat long
## sqft_basement 0 0 0alias has detected that sqft_basement can be calculated from sqft_living minus sqft_above. This means we can removed sqft_basement.
houses_tidy <- houses_tidy %>%
select(-sqft_basement)Question 3
houses_tidy_numeric <- houses_tidy %>%
select_if(is.numeric)
houses_tidy_nonnumeric <- houses_tidy %>%
select_if(function(x) !is.numeric(x))
houses_tidy_nonnumeric$price <- houses_tidy$priceggpairs(houses_tidy_numeric, progress = FALSE)
Correlation with price - Numeric Variables
Strong positive correlations 1. sqft_living (0.702) 2. sqft_above (0.606)
Moderate correlation 3. bathrooms (0.525)
Weak correlations 4. bedrooms (0.308) 5. latitude (0.307) 6. floors (0.257)
Very week correlations 7. sqft_lot (0.09) 8. yr_built (0.054) 9. longitude (0.022)
ggpairs(houses_tidy_nonnumeric, progress = FALSE)
Correlation with price - Non-Numeric Variables
waterfront - appears to affect price
View - there is a correlation with price, the median of the 5 levels generally increases with better views (less obvious between levels 1 and 2). The highest and lowest levels of view clearly have an effect on price.
condition - no obvious correlation here
grade - looks like a good correlation here with a decent increase in price related to the highest grade of building. This looks like the strongest correlation from the non-numeric variables.
renovated - may be a correlation but not a strong one from looking at boxplots
Conclusion predictors to investigate in order of potential grade = “high” waterfront = TRUE view = 4 (or view - 0, I’m not sure)
First Predictor - sqft_living
model1 <- lm(price ~ sqft_living,
data = houses_tidy)
summary(model1)##
## Call:
## lm(formula = price ~ sqft_living, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1476062 -147486 -24043 106182 4362067
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -43580.743 4402.690 -9.899 <2e-16 ***
## sqft_living 280.624 1.936 144.920 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 261500 on 21611 degrees of freedom
## Multiple R-squared: 0.4929, Adjusted R-squared: 0.4928
## F-statistic: 2.1e+04 on 1 and 21611 DF, p-value: < 2.2e-16autoplot(model1)
plot(model1)
Not sure these diagnostic plots look great but have plotted the data below and I think the issue is caused by the shape of the data. It appears to have very defined cut-offs but I don’t know why?
houses %>%
ggplot(aes(price, sqft_living)) +
geom_point(alpha = 0.1)
Conclusion I have reservations about the shape of the data and consquently the diagnostic plots but lets go with this for now. R2 = 0.493 rse = 261500 (this is really quite high!) p-value < 0.01
Second Predictor
The next strongest correlation for the numeric variables was sqft_above but this is really very similar to sqft_living so lets not do it next. Bathrooms has a moderate correlation with price so should have tried that next but in my exhausted delirium I added bedrooms by accident. Realise when I came back to check things over but when I changed bedrooms to bathrooms I found it didn’t improved the model at all so maybe bedrooms is the better chose as 2nd predictor.
model2 <- lm(price ~ sqft_living + bedrooms,
data = houses_tidy)
summary(model2)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1650867 -143866 -23143 102344 4179850
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 79469.359 6604.764 12.03 <2e-16 ***
## sqft_living 313.949 2.337 134.31 <2e-16 ***
## bedrooms -57066.759 2308.223 -24.72 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 257800 on 21610 degrees of freedom
## Multiple R-squared: 0.5068, Adjusted R-squared: 0.5068
## F-statistic: 1.11e+04 on 2 and 21610 DF, p-value: < 2.2e-16R2 = 0.508 so not added much from model 1 (R2 = 0.493) rse = 257500 still very high
autoplot(model2)
plot(model2)
We seem to be fitting a negative values - what is going on here?
houses %>%
ggplot(aes(price, bedrooms)) +
geom_point()
So there is a house with over 30 bedrooms. This was investigated. This house is noted as having 33 bedrooms but costing $640000. All the other houses that cost this much (there were 21) had between 2 and 6 bedrooms, this is therefore assumed to be a typo and the number of bedrooms changed to 3.
houses_tidy <- houses_tidy %>%
mutate(bedrooms = ifelse(bedrooms > 30, 3, bedrooms)) Run model and diagnostic again…
model2b <- lm(price ~ sqft_living + bedrooms,
data = houses_tidy)
summary(model2b)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1666625 -143358 -23058 102392 4163512
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 90059.793 6733.139 13.38 <2e-16 ***
## sqft_living 316.914 2.365 134.00 <2e-16 ***
## bedrooms -62063.528 2392.305 -25.94 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 257500 on 21610 degrees of freedom
## Multiple R-squared: 0.5082, Adjusted R-squared: 0.5081
## F-statistic: 1.116e+04 on 2 and 21610 DF, p-value: < 2.2e-16The R2 and rse are similar to model2 but the diagnostics should look better.
autoplot(model2b)
Hmmm, for plot 2 the residuals are getting larger as the prices get higher. For scale-location, the blue line is not flat and I think it should be. Am starting to have my doubts about using sqft_livng, there is something not right with this data.
Does sqft_above look any better?
houses_tidy %>%
ggplot(aes(price, sqft_above)) +
geom_point()
No the sqft_above data looks the same as sqft_living
What if we just try bedrooms on its own as the first predictor?
First predictor (again!)
model1b <- lm(price ~ bedrooms,
data = houses_tidy)
summary(model1b)##
## Call:
## lm(formula = price ~ bedrooms, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -993323 -203016 -65422 105984 6824400
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 110333 9108 12.11 <2e-16 ***
## bedrooms 127544 2610 48.87 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 348400 on 21611 degrees of freedom
## Multiple R-squared: 0.0995, Adjusted R-squared: 0.09946
## F-statistic: 2388 on 1 and 21611 DF, p-value: < 2.2e-16R2 is low, bedrooms isn’t explaining much of the variation. I think we have to go with sqft_living.
Lets press on with model2b (price ~ sqft_living + bedrooms(with 33 altered))
Third Predictor
model3 <- lm(price ~ sqft_living + bedrooms + waterfront,
data = houses_tidy)
summary(model3)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms + waterfront, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1547815 -139924 -20327 103317 4271814
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 83206.467 6497.811 12.80 <2e-16 ***
## sqft_living 304.569 2.302 132.29 <2e-16 ***
## bedrooms -54179.832 2316.240 -23.39 <2e-16 ***
## waterfrontTRUE 790888.877 19706.699 40.13 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 248400 on 21609 degrees of freedom
## Multiple R-squared: 0.5423, Adjusted R-squared: 0.5422
## F-statistic: 8534 on 3 and 21609 DF, p-value: < 2.2e-16We’re up to R2 = 0.542 now, rse reducing slightly but still very high. p-values very low and diagnostic plots ok
autoplot(model3)
We’re up to R2 = 0.542 now, rse reducing slightly but still very high. p-values very low and diagnostic plots ok (ish) (I think??) Something odd has happened to leverage graph
Fourth Predictor
Lets look at the residuals to see what can best explain them.
houses_resid <- houses_tidy %>%
add_residuals(model3)houses_tidy_numeric <- houses_resid %>%
select_if(is.numeric)
houses_tidy_nonnumeric <- houses_resid %>%
select_if(function(x) !is.numeric(x))
houses_tidy_nonnumeric$resid <- houses_resid$residggpairs(houses_tidy_numeric, progress = FALSE)
Correlation with Residuals - Numeric Columns
latitude - weak positive (0.396) close to being moderate though yr_built - weak negative (-0.240)
ggpairs(houses_tidy_nonnumeric, progress = FALSE)
Correlation with Residuals - Non-Numeric Columns
grade = “high” is showing a bit of correlation None of the others are very convincing
Add grade as 4th predictor
model4 <- lm(price ~ sqft_living + bedrooms + waterfront + grade,
data = houses_tidy)
summary(model4)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms + waterfront + grade,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1583426 -136185 -22248 99412 4189856
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 82809.000 120395.386 0.688 0.492
## sqft_living 264.470 2.477 106.770 < 2e-16 ***
## bedrooms -40777.877 2274.891 -17.925 < 2e-16 ***
## waterfrontTRUE 759150.255 19121.073 39.702 < 2e-16 ***
## gradeaverage 28275.650 120492.322 0.235 0.814
## gradehigh 485138.645 121159.406 4.004 6.25e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 240800 on 21607 degrees of freedom
## Multiple R-squared: 0.57, Adjusted R-squared: 0.5699
## F-statistic: 5728 on 5 and 21607 DF, p-value: < 2.2e-16R2 is now 0.57 with rse 240800 grade = “high” has a p-value < 0.01 but grade = “average” has a high p-value and is not significant.
autoplot(model4)
plot(model4)
anova(model3, model4)Adding grade is statistically significant
anova(model2b, model3)Adding waterfront is statistically significant.
Third Predictor (again)
What if we try latitude as 3rd predictor?
model3b <- lm(price ~ sqft_living + bedrooms + lat,
data = houses_tidy)
summary(model3b)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms + lat, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1657228 -121478 -19654 80401 4191985
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.333e+07 5.584e+05 -59.69 <2e-16 ***
## sqft_living 3.073e+02 2.196e+00 139.93 <2e-16 ***
## bedrooms -5.520e+04 2.219e+03 -24.88 <2e-16 ***
## lat 7.026e+05 1.174e+04 59.85 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 238500 on 21609 degrees of freedom
## Multiple R-squared: 0.5781, Adjusted R-squared: 0.578
## F-statistic: 9870 on 3 and 21609 DF, p-value: < 2.2e-16Adding latitude as 3rd predictor takes us up to R2 = 0.5781
Fourth Predictor (again)
Lets add previous 3rd predictor in which was waterfront
model4b <- lm(price ~ sqft_living + bedrooms + lat + waterfront,
data = houses_tidy)
summary(model4b)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms + lat + waterfront,
## data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1534034 -116206 -16352 81061 4304633
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.392e+07 5.338e+05 -63.55 <2e-16 ***
## sqft_living 2.944e+02 2.118e+00 138.96 <2e-16 ***
## bedrooms -4.691e+04 2.128e+03 -22.04 <2e-16 ***
## lat 7.150e+05 1.122e+04 63.71 <2e-16 ***
## waterfrontTRUE 8.190e+05 1.809e+04 45.28 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 227900 on 21608 degrees of freedom
## Multiple R-squared: 0.6147, Adjusted R-squared: 0.6146
## F-statistic: 8617 on 4 and 21608 DF, p-value: < 2.2e-16Cool, now we’re at R2 0.6147 rse still horribly high p-values all very low
autoplot(model4b)
Residuals
Lets look at the residuals again to see what can best explain them.
houses_resid <- houses_tidy %>%
add_residuals(model4b)houses_tidy_numeric <- houses_resid %>%
select_if(is.numeric)
houses_tidy_nonnumeric <- houses_resid %>%
select_if(function(x) !is.numeric(x))
houses_tidy_nonnumeric$resid <- houses_resid$residggpairs(houses_tidy_numeric, progress = FALSE)
yr_built (-0.188) and longitude (-0.153) both have very weak negative correlations with the residuals
ggpairs(houses_tidy_nonnumeric, progress = FALSE)
Could try view? Grade doesn’t seem to make sense anymore as the lowest grade has a higher median than average and isn’t much different from the high level?
Seventh Predictor!!!
What happens if we add all our remaining potential predictors?? Woohoo, am going nuts!
model5 <- lm(price ~ sqft_living + bedrooms + lat + waterfront + long + view + grade,
data = houses_tidy)
summary(model5)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms + lat + waterfront +
## long + view + grade, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1762508 -104948 -11871 76237 4116519
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.696e+07 1.350e+06 -42.199 <2e-16 ***
## sqft_living 2.496e+02 2.324e+00 107.401 <2e-16 ***
## bedrooms -3.027e+04 2.022e+03 -14.971 <2e-16 ***
## lat 6.761e+05 1.062e+04 63.667 <2e-16 ***
## waterfrontTRUE 4.949e+05 2.079e+04 23.803 <2e-16 ***
## long -2.051e+05 1.090e+04 -18.814 <2e-16 ***
## view1 1.289e+05 1.187e+04 10.856 <2e-16 ***
## view2 9.913e+04 7.151e+03 13.863 <2e-16 ***
## view3 1.724e+05 9.787e+03 17.618 <2e-16 ***
## view4 3.335e+05 1.513e+04 22.037 <2e-16 ***
## gradeaverage -1.755e+05 1.066e+05 -1.646 0.0997 .
## gradehigh 2.463e+05 1.072e+05 2.298 0.0216 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 212800 on 21601 degrees of freedom
## Multiple R-squared: 0.6641, Adjusted R-squared: 0.664
## F-statistic: 3883 on 11 and 21601 DF, p-value: < 2.2e-16Still only at R2 = 0.664 rse = 212800 so still very big
The grades aren’t statistically significant which isn’t that surprising. Lets change it to condition.
model5b <- lm(price ~ sqft_living + bedrooms + lat + waterfront + long + view + condition,
data = houses_tidy)
summary(model5b)##
## Call:
## lm(formula = price ~ sqft_living + bedrooms + lat + waterfront +
## long + view + condition, data = houses_tidy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1752924 -108306 -10720 82648 4144185
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.723e+07 1.391e+06 -41.150 < 2e-16 ***
## sqft_living 2.897e+02 2.196e+00 131.910 < 2e-16 ***
## bedrooms -4.519e+04 2.058e+03 -21.953 < 2e-16 ***
## lat 6.847e+05 1.093e+04 62.629 < 2e-16 ***
## waterfrontTRUE 5.060e+05 2.137e+04 23.680 < 2e-16 ***
## long -2.021e+05 1.124e+04 -17.980 < 2e-16 ***
## view1 1.236e+05 1.221e+04 10.127 < 2e-16 ***
## view2 9.910e+04 7.353e+03 13.477 < 2e-16 ***
## view3 1.742e+05 1.006e+04 17.311 < 2e-16 ***
## view4 3.451e+05 1.556e+04 22.184 < 2e-16 ***
## condition2 3.018e+04 4.329e+04 0.697 0.48569
## condition3 2.548e+04 4.002e+04 0.637 0.52442
## condition4 6.352e+04 4.008e+04 1.585 0.11302
## condition5 1.089e+05 4.033e+04 2.700 0.00693 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 218700 on 21599 degrees of freedom
## Multiple R-squared: 0.6453, Adjusted R-squared: 0.6451
## F-statistic: 3022 on 13 and 21599 DF, p-value: < 2.2e-16Only condition = 5 is statistically significant. I think this means I have to group them?
Might have to give up here and go to bed. Night night zzzzzzzzz